Hi Steve. There are two thermostats clamped to the top of the boiler. In normal (coffee) mode the coffee thermostat opens once the machine reaches the proper temperature, this illuminates the ready light and cuts power to the heating element. As the machine cools the thermostat "closes" again, turning off the light and restoring power to the heating element. If the steam button is pressed this bypasses the coffee thermostat and instead uses the steam thermostat to control heating and the ready light. It's clear from the described symptoms that your coffee thermostat has failed in the "open" position, and you can verify this by checking it's continuity with a voltmeter. This guide will walk you through that; the coffee thermostat should be the one closer to the front of the machine, and if it's "open" with the machine cool then it has failed and needs to be replaced:
Testing the Starbucks Barista Heating Element
The part number for the coffee thermostat is 189422000 and it can be purchased here:
Best of luck with the repair!